Addendum to "Ultraviolet behaviour of Higgs inflation models"

This article is an addendum to [1]. We extend our computation of the on-shell scattering amplitudes in an arbitrary inflaton background $\bar{\phi}_1$. Although the effective Einstein frame cutoff for $\bar{\phi}_1>>M_P/\sqrt{\xi}$ turns out to be $\bar{\phi}_1$ or $\xi\bar{\phi}_1$ for the $U(1)$ model, this is not the case for the realistic doublet Higgs model where the effective Einstein frame cutoff turns out to be the standard $M_P/\sqrt{\xi}$ for both the Palatini and metric formulations. Then, as it has been pointed out in [1] the background $\bar{\phi}_1$ is the effective Jordan frame cutoff for both the Palatini and metric formulations.